CAIE M1 (Mechanics 1) 2004 November

1
\includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-2_200_529_269_808} Two particles \(P\) and \(Q\), of masses 1.7 kg and 0.3 kg respectively, are connected by a light inextensible string. \(P\) is held on a smooth horizontal table with the string taut and passing over a small smooth pulley fixed at the edge of the table. \(Q\) is at rest vertically below the pulley. \(P\) is released. Find the acceleration of the particles and the tension in the string.

2 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A small block of weight 18 N is held at rest on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal, by a force of magnitude \(P\) N. Find

1. the value of \(P\) when the force is parallel to the plane, as in Fig. 1,
2. the value of \(P\) when the force is horizontal, as in Fig. 2.

3 A car of mass 1250 kg travels down a straight hill with the engine working at a power of 22 kW . The hill is inclined at \(3 ^ { \circ }\) to the horizontal and the resistance to motion of the car is 1130 N . Find the speed of the car at an instant when its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

4 A lorry of mass 16000 kg climbs from the bottom to the top of a straight hill of length 1000 m at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 20 m above the level of the bottom of the hill. The driving force of the lorry is constant and equal to 5000 N . Find

1. the gain in gravitational potential energy of the lorry,
2. the work done by the driving force,
3. the work done against the force resisting the motion of the lorry. On reaching the top of the hill the lorry continues along a straight horizontal road against a constant resistance of 1500 N . The driving force of the lorry is not now constant, and the speed of the lorry increases from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(P\). The distance of \(P\) from the top of the hill is 2000 m .
4. Find the work done by the driving force of the lorry while the lorry travels from the top of the hill to \(P\).

5
\includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-3_240_862_274_644} Particles \(P\) and \(Q\) start from points \(A\) and \(B\) respectively, at the same instant, and move towards each other in a horizontal straight line. The initial speeds of \(P\) and \(Q\) are \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The accelerations of \(P\) and \(Q\) are constant and equal to \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively (see diagram).

1. Find the speed of \(P\) at the instant when the speed of \(P\) is 1.8 times the speed of \(Q\).
2. Given that \(A B = 51 \mathrm {~m}\), find the time taken from the start until \(P\) and \(Q\) meet.

6
\includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-3_330_572_1037_788} Two identical boxes, each of mass 400 kg , are at rest, with one on top of the other, on horizontal ground. A horizontal force of magnitude \(P\) newtons is applied to the lower box (see diagram). The coefficient of friction between the lower box and the ground is 0.75 and the coefficient of friction between the two boxes is 0.4 .

1. Show that the boxes will remain at rest if \(P \leqslant 6000\). The boxes start to move with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that no sliding takes place between the boxes, show that \(a \leqslant 4\) and deduce the maximum possible value of \(P\).

7 A particle starts from rest at the point \(A\) and travels in a straight line until it reaches the point \(B\). The velocity of the particle \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.009 t ^ { 2 } - 0.0001 t ^ { 3 }\). Given that the velocity of the particle when it reaches \(B\) is zero, find

1. the time taken for the particle to travel from \(A\) to \(B\),
2. the distance \(A B\),
3. the maximum velocity of the particle.